The tangent line to the graph of function $g$ at the point $(9,2)$ passes through the point $(5,7)$. Find $g'(9)$. $g'(9)=$
Answer: The derivative of a function at a point gives the slope of the line tangent to the function's graph at that point. Therefore, $g'(9)$ gives the slope of the tangent line to the graph of $g$ where $x=9$, which is the point $(9,2)$. We know this line passes through $(9,2)$, and we are also given that it passes through $(5,7)$. This should be enough to find the slope of that line. $\begin{aligned} \text{Slope}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{7-2}{5-9} \\\\ &=\dfrac{5}{-4} \end{aligned}$ In conclusion, $g'(9)=-\dfrac{5}{4}$.